Complex biological systems are increasingly subject to investigation by mathematical modeling in general and stochastic simulation in particular. Advanced mathematical methods will be used to generate next-generation computational methods and algorithms for (1) formulating these models, (2) simulating or sampling their stochastic dynamics, (3) reducing them to simpler approximating models for use in multiscale simulation, and (4) optimizing their unknown or partly known parameters to fit observed behaviors and/or measurements. The proposed methods are based on advances in applied statistical and stochastic mathematics, including advances arising from operator algebra, quantum field theory, stochastic processes, statistical physics, machine learning, and related mathematically grounded fields. A central technique in this work will be the use of the operator algebra formulation of the chemical master equation. [unreadable] [unreadable] The biological systems to be studied include and are representative of high-value biomedical target systems whose complexity and spatiotemporal scale requires improved mathematical and computational methods, to obtain the scientific understanding underlying future medical intervention. Cancer research is broadly engaged in signal transduction systems and complexes with feedback, for which the yeast Ste5 MARK pathway is a model system. DNA damage sensing (through ATM) and repair control (though p53 and Mdm2) are at least equally important to cancer research owing to the central role that failure of these systems play in many cancers. The dendritic spine synapse system is central to neuroplasticity and therefore human learning and memory. It is critical to understand this neurobiological system well enough to protect it against neurodegenerative diseases and environmental insults. The project seeks fundamental mathematical breakthroughs in stochastic and multiscale modeling that will enable the scientific understanding of these complex systems necessary to create effective medical interventions of the future. [unreadable] [unreadable] [unreadable]